Elem Particle Theory - refs & notes

Elementary Particle Theory Various Refs&notes ; PhilLucht 1979 O01.GesWig ST Se S79 Ttbaplehkved48)Shuler wif GaaFagSLR ateinLEATaa Osu BpMEaesosmeee, 1) Byorcasve hyaeai oO ThyRS eeBRE “6t)Sueal2aals eonspret 24)‘eeeClllePansianyreanenenegw One cow ten )—_yery(onJaahiet)seatingaleSess 22)Dee. ; 2)ahpas[Lis-coesh—Shane)slaGE)aovatAFNN 2e)WomanRiva = (ACen) EDsdbfP yor | CPeet po)taerea)~34,Nonesghwinery! -—Malar'yahors qeomy \ ; oudn u me wuecha a5)Lectura eas Oesevens 32)SeeTxreview4)remem4NAGY. ~ -)Ce te ‘ oe : Volume 658,number 4 PHYSICS LETTERS 6December 1976 ae : BEHAVIOUROFLEADINGINFRA-REDDIVERGENCES E OFQCD INTHE AXIAL GAUGE ze. E:- J.FRENKEL* Pe Departmen ofTheoretical Physics,UniversityofOxford,Oxford,UK i Received 16September 1976 * Weidentity inthe axial gauge asimpic sotofdiagrams which yield theexponential ofleading infra-ed corrections EY tomassive quatk scattering inQCD. The relevance oftheWard identities ispointed out. , Recently itwasshown [1]thattheleading infra-red divergent corrections tomassive quarkscattering byanex- t ternal colourless potential inQCD areverysimple tofourth order, bothforexclusive andinclusive reactions. This :analysiswascarriedoutintheFeynmangaugeandusehasbeenmadeofdimensional regularization with space- timedimension 4+e.Intheregime when themomentum transfer |qlisofthesame order ofmagnitude asthe |! ‘quark mass mitwasfound that, tothisorder, thescattering amplitude obeys theequation: a 7 7 575002, CeNGAEOS(A?, @ Gpisthequadratic Casimir operator inthequarkrepresentation and ; 1(1+Ltr) rs ayn (Le?der re 2/q2)-Md ® wey(YE miei), raceamtety @ z Fortheexclusivereaction,withwhichwewillbeconcernedhere,¢=~€~!,Finally#(0)istheeffectivecoupling si constantofthePureYang~Millstheoryasdefinedbytherenormalization group.Jntheabovereference,aswell _asinref.[2],severalreasonsweregivenwhichsuggestthateq.(1)mightbetruetoallorders.Inthiscaseitim: $ipliestheexponentiation ofleading infrared divergences a Since intheaxialgauge (3],(0)isclosely connected withtheself-energy correction tothegluon propagator, a itisnatural totrytounderstand eq.(1)inthisgauge # ra!Inthisnotewewouldliketoexhibitpreciselythesetofdiagrams which yield eq.(I)inthisgauge. Tothisend, ' wweshall firstillustrate tothe4thorder, therelevance oftheWard identities. Consider fordefiniteness, theself ‘energy insertion onthequark legofmomentum pshown infig.1\Graphs containing crossed gluon lineshavebeenomittedsincetheyarenon-leading inourregime[4]}Alsoonlythegraphscontsibuting totheoneparticle §ineducible self-energy E*havebeenexplicitly shown.Thereasonisthatthecompletecontribution isgivenbytheinverse ofthefunction Cwhich isdetermined bythebehaviour ofE*near mass shell: E*(p,n) =iC, n)(ig+m), 6) . where denotes ageneral fixed fourvector specifying thegauge. Thefunction Cisobtained bydifferentiating 2°withrespecttothemomentuin andtakingtheon-shellmateixelementsoftheresultingexpression.Inthisway, 3 C(p,n)is given to4thorder bythesetofdiagrams shown infig.2,where only thediagrams containing leading divergences have been explicitly shown. Byacompletely similar procedure wealsoobtain C(p', n).Furthermore, considering gluon selfenergy aswell asfermion vertex andself-energy corrections totheskeleton diagram, one *Oneave ofabsence from InstitutodeFisica,UniversidadedeSoPaulo,Brasil.SupportedinpartbyFundacfodeAmparo Pesquisa doEstado deSo Pavlo+1Thecouwinisnce ofthisgaugehasalsobeenpointedoutinthisconncetion byCoonwalinef(2]- 383 | I Maes i osoo = a Asymptotically FreeGauge Theories. I* David J.Grosstt FrankWilezek d ‘ 5 tanceduetotherecentdiscoverythatnon-Abelian (wyorAg")couldbeneglected.Inotherwords iqgaugetheorles areasymptotically free. Inthis theleading asymptotic behavior oftheGreen’s portedinRef.1 n'anasslesa theory, Thiscanbeproved, toany ¢Therenormalization groupdatesfromthefunda- finiteorderinperturbation theory,byusing §mentalworkofGell-Mann andLow,*whostudied Weinberg’s theorem. Themassless theorycon- & inquantum electrodynamics. Theremarkable ofmomenta, therefore onemightexpectthatthe t ymptotic formofthephotonpropagator wasdeter- determined bypuredimensional analysis. Thisis PF i trodynamics andotherfieldtheories.*” (Fora singularities thesesubtractions, forthemassless 7‘ review ofthisworkseeRef.8.) theory, mustbeperformed offshell, sayatsome fieldtheory contains twotypesofparameters— scateofthefields (which aredetermined bythemassesorcoupling constants withpositivedimen- ‘wave-function renormalization constants). The {sionsofmass(i.e.,dueto\/gporAg?termsin subtraction point,y,isarbitrary. If'wechange if stants (.e.,ductoag*orGA"pA,terms inthe valueofthecoupting constants andthescaleofthe 8 3633‘ Derivation ofthe Renorm group equations. This isthe best discussion Ihave seen anywhere. Weare talking pure Yang Mills intheLorents gauge with ghosts andageuge parameter a.Thebasic numbers Z,, 25zy,%,areclearly defined injustthewayIwouldhavedoneit.Therenorm point isso intheEuclidean region tokeepawayfromnormal thresholds etc. Ifyouchange ufromonevalue toanother Iwould expect alltheZ'stochange. Bytheway,theZ'sherearefunctions ofg,y%4sA.ptsIe,youcompute your unrencrned vertex, say, tosome order ing, with acutoff. Abarbanel does allthis stuff in the dim reg. SotheZ'saredefined thus. Whyareyouallowed, eg,toinsist on(2.8)? This represents thevalue ofthe“constant” term inthevertex. Ie,ifyouwere toTaylor expand aboutthethreea”=eh thezeroth orderterminyourexpansion isjust the RHS of(2.8). Buthowdoyouknow,youmightask,thatitisproportional tothebarevertex? /Here isthe argument: imagine expanding inthetriple Taylor series just mentioned, Iknow that ifIperform asingle subtraction atthe renorm point, the result will befinite. Thus, Iknow that the subtraction process removes precisely this first term. Onthe other hand, Iknow that subbraction means adding acounterterm tothe (1agrangian ofthesameformasthecoupling already there. Thecoupling thatisalready ysSS,thereisthebarevertex", thusasubtraction canonlyremoveamultiple ofthebarepo" vertex. SinceYMTHisrenormelizesble, youtherefore knowthatthatfirstconstant jtermintheToylor series isamultiple ofthebarevertex! Weweresurprised when'wefound this bydirect calculation, but ofcourse weshould not have been surprised. Similarly, weexpect the propagator attherenorm point tobeproportional to thebarepropagator, andthesamefortheghostfunctions. Ifthiswerenotthe | case,youcouldnotrenormalize bydoingthesubtractions inplied bycounterterms. (So, inconclusion, weknow that the Z's all exsit and can becomputed to any order, aslong asyou have acutoff oradim reg procedure. And changing the renorm point should change the Z's. You cannot just choose theZ's tobewhat you want, not at this stage. What happens next? Theusual multiplicative renorm program isdone. Youdefine renormed fields, coupler, andgauge parameter, and1PIGreens functions. Byrealizing that theunrenormed Green Functions “never heard of m"yougetthefamouy R-group diefferential equation (2.17), seedetail attached. 7 QTR Fp aceaeooMamieDae—Ope9BsCen —SSR) OMA uewoe? lewbade meeos Del(adlyse _ Frew SEEatcontd ddOakRean)=(21)=0d(Oo rene [Azesa ae —_— RS EEE Nnfad, eem nsz. 4 watchCES. -@Lawacoor \dooomasnia, oadDarSoVib,cueFoufee(ON — duede=LtiteOSCos=mCR).<TheOraCau, ee eg te ©.Qos00kSeahvonDttind.OegetSwing8D_ousytoounces, i, owGalatceedeeeattegaOokbeg ° Asymptotic Gauge Theories I Gross and Wilczek rec 7423.73 pub PRDS,3633 (1973) 1.Introduction, Wehave here ahistorical review ofrelated work. Ihave taken notes inrefssection, thefirst30refsareinvolved. Thebigrecentpushofcourseisthe discovery byauthors that NAGT possess the quality ofasymptotic freedom. Old,work was done byGell_Mann Low and Bogo Shirkov group inthe renorm group equations, &references 4thru 8.These guys wrote down RGequations and realized that your subtraction point changes your charges etc, butthework sayidle formany years. Reason: you couldn't calculate anything. Eg, GLcould not compute the asymptotic behavior ofQED, though they new about invariant charges etc. Another reason was that RGE connected you todeep Buclidean region ofgreens, not ofphysical interest asin S-Matrix elements, sonoone really cared. Asanaside, during this dormant period Johnson Baker Wiley developed their finite QED ideas using R-group but not insame way asused here. Then came the experimental work indeep indlastic and various theories to trytoexplain scaling.: Partons, light cone, current commutators, andsoon. Eg,Bjorken sealing was coindd in1969. Fgynman's book reviews this era. Along with this experimental stuff came Callan Symanzik scaling theory and Wikson operator product theory ,allrealted toRegroup. Everything seemed tofittogether ifonly youhadaUV~stable fixed point atZERO! Atheory with this property wascalled asymptotically free, .In1973, Zee showéd that scalr QED was not AF. Authors showed that you had tohave non-abélian gauge particles tostand achance. Then in1973@©wasfound apositive proof: that NAGT were AF.Ofcourse gauge theories were useful from other viewpoints: field theories, renormalizeability had been proved in1971, Weinbergs theory lookdd good forweaks, SSB, etcetc. Now itlooked like the deep inelastic partonic results could beconnected in!!! Intro ends with anoutline ofpaper. Pmog FF a VARIATIONAL UPPER BOUND ONTHE SCATTERING LENGTH 933, ‘ $56t:2)E.¢. we=. From(4.6)and(2.4),itfollowsthat(C)=-a. fdF06,F;18G,3;0-1)=88,3;0), Sinceithasalready beenshownthatg(B)=Ay, and weobtain . since itisclear from Eq. (2.4) that g(B+C)=(B) rencet. +8(C),weseefromEq.(3.4)that caf4596,3;ane). an 7Kre 422)=A,-a. From the relation® apo-shltin =y12 InviewofEq.(4.3)andtherelation BAF,F;8)~(2npt*/m)-*/*|9(0, F)|,Be(4.8) a-=, H§j(0,7)=0, with(0,#)denoting thezero-energy limitofthe 0, F)=0 outgoing scattering solution j(k,F)oftheSchré- theVUBonA,canalsobewritten asdingerequation(Ey~#)§=0,itthenfollowsthat m=A, IF§*(0,F)HG(O, F) . C~(228h*/m)"Ha, po (4.8) KEDAsrs TA[THODMOD, (8) aa fdF5*(0,FWH)HL0,2). (4.7) whichisthedesiredresult 'D.Gelman andL.Spruck, J.Math. Phys.10,2240 "derived inRef.1,which feapplicable alsotothebound(068). obtainedherein,iagivenintheIntroduction ofRef.1.2R.P.Feynman, Rev.Mod.Phys.20,367(1948). 4s,Servadio, Phys. Rev.A4,1256(1971).AL,Spruch andL.Rosenberg, Phys, Rev.116,1034 "Thisrelation follows fromtheequation J(s)=J"(s),(959). whereJandJ"aredefinedbyEs.(D1)and(D2)of‘Allofthetermsandnotationswhichweareusingwere. Ref.1,bythomothodexactlyanalogoustothatusedIndefinedinRet.1;theywillnotborepeatedhero. derivingEa,(Dl4)fromBq,(019).*Abrietdiscussion ofthepotential utility ofthebounds "This follows from Eqs.(2.8)and(2.4)ofRef.1, PHYSICAL REVIEW D VOLUME 9,NUMBER 4 15PEBRUARY 1074 Renormalization ofgauge theories—unbroken andbroken Benjamin W, LeetNationalAccelerator LaboratsryBatovie, Mixots60610{Received 29Oetober 1972) : ‘Acomprohensive discusston fsgiven oftherenormalization ofgauge theories, with or without spontaneous beeakown ofgauge eymmctry. Thepresent discussion makes useof theWard-Takahashi identities forproper vertices (asopposed totheidentities forGreen's funetions) recently derived. Thefollowing features oftnopresent discussion aresignificant: (0)Thepresent discussion applies toavery wide class ofgauge conditions.) Thepresent Ascussion applies toanygauge group andanyrepresentation ofthescelae fields.) The renormatized §matrix isshown tobegauge-independent. (4)Dependence ofcounterterms onthegauge chosen isdiscussed, LINTRODUCTION Inthis paper weshail reexamine therenormal- tzability ofgauge theories interms oftheWTideri~ Inanearlier publication,! wehave given ader- tities forproper vertices. Inaddition forederiv- ivation oftheWard-Takahashi (WT) identity for ingmany results ofRefs. 2,3,and5(which we thegenerating functional ofproper vertices innon- _shall refer toasL.ZI, L2Il, andLZIV, respective- : Abelian gauge theories. Previous discussions on ly),weshall addthefollowing elements toourdis- therenormalizability ofgauge theories*-* were cussions: (1)Weshall discuss therenormalizabili- . based ontheWard-Takahashlidentities forGreen's tyofgaugetheoriesinawideclassofgaugecon- functions.*:7Therenormalization procedure is ditions. Thegaugeconditions weshallconsider : Usuallystated intermsofproper vertices, sothat arelinear infieldvariables andofdimension 2or : theWTidentitiesforproperverticeswouldfacili- ‘less.Mostgaugeconditions considered inthelit- tateenormously thediscussion ofrenormalizability. erature**“ areofthiskind.(2)Weshallconsider ! ~ oe Zo ahF,Bradainanteetal,Highenergyhadron-deuteron scattering saa NucleutPhysis833(1971)173-199,NortelPublishingCompany aqaft [121 Re. cho, Cotanan E7. Rhones, M,Flings, P-Gusman,RE,Lamb,FC.Ped °7"gonandLS.Schrooder,Phys.Rev.Lett22(1969)1265. ag U3]J.L,Feedes 8.7,Emerton, H,Palesky, W.D.Simpeon, RU.Suter, Rul,Stearns and: WW.Von Witsch, Phys, Rev, Lett 24(1970) 677. ° 14]G.AltieriandL.fertocehi, NuovoCim,634(1969)285. 4 {is]VearrandRA.PilijnPyeLet260(1968)730 > RENORMALIZATION OFMASSLESS (16]V.BargerandRJ.N.Phillips,Phys.Rev.Lett.21(1968)865. ry YANG-MILLS FIELDS . [13] C.MichaelandC.Witkin,Nucl.Phys.B11(1969)99, a [18]¥.BargerandRN.Philips,Phys.Rev.187(1969)2210, oo *tHOOFT . [19] K.3, Foley, S.J. Lindenbaum,W.A.Love,S,Ozaki,J.RussellandL.C.L.Yuan,Phys h GALHOOFT: IntttuteforTheoretiealPhysies.UnversityofUtrecht Rev.Lett.11(1963)425, a orePhi.Un 20]KJ.Foley,R.S.Gilmore8.1,Lindenboum, W.A.Love,S.Ozaki,E.H.Willen,R.Yai mecht,TheNetherlands andLCLL. Yuan, Phys. Rev.Let, 19(1967) 857. ty [21]D.Harting, P.Blackall, B.Elsner, A.C.lolmhole, W.C,Middetkoop, B,Powel, B,Zac Received 12February 1971P.Zanella, P.Dalphiaz, MLN.Focacci, S.Focardi, G,Giacomell, L.Monari, JA.Beasey .R.A,Donald, P.Mason, LW.Jones andD.O.Caldwell, Nuovo Cin38(1963) 60.-¢K.Horn,J.Kénig,F.Ménnig,P.Schludecker, I.Sehopper,P.Sievers,H.Ue - aanaangePipeLat35D9G) neeTESehones NE"|worceTeproblemofroxmaliation ofgaugeldssted.Iseedthathewe spcsaplacadARecsPhosLaveoniS ofnon-gaugeinvariantregulatorfieldsisnotexcludedprovidedthatinthelimitofhigh {23)1PapinandMt,Ros,PhysLet,21(1568)1778. wh cruevaratocedprovehatnhiitofig 124}G,Alber!andL.Bertocehi,NuovoCim,614(1969)203. ye ‘enerenueiraecobeeredbyreason numerofcounter (25)G.Fate,Recoileffetinhihreneryhadson-deuteron colisions,HarvardUnivesityprtemsinheLagrangian,MasYangilseldscanbeustedathanne print,1970,. fie|""Consiency oftnemethod'sprovedfordsgrmswithnonoverlapping divergences 26)W.F.Baker,E,W,Jenkins,TF.Kycia,Rl,Philips,A.L.Read,K.P.RileyandH.RudeJ. yoftha Provedfordig pingdivers 1ea ep, Pro Int Conon clamentry partesanahigioneeypyien, Sionna(19635 bymeansofgaugeinvasiantregulators,whichhowever,cannotbeinterpretedintermsof (eds.G,BernardiniandG.P,PuppinSocietsHalianadiFisica,Bologna,1963),volI repslitorfields.AssumingconsistencytheSomatixisshowntobeunitaryinanyorder {eassxE-] ofthecouplingconstant. Arestriction mustbemade:nolocal.paity-changing trnator= 27]W,Gatbraith,E.W.Jenkins.TF.Kyeia,B.A.Leonté,RllPhillips,A.L,ReadandReRrmationsmustbecontainedintheunderlyinggaugegroup,Theinteractions mustcon- binstein,Phys,Rev,1388(1965)913. iH sevepatty 128] F.Bradamamte, S.Coneti,G, Fidecaro, M,Pidccaro, M.Giorgi, A.Penzo, L.PlemoiiéxandSchiavon,Proc.oftheTopicalSeminaroninteractions ofelementary particleswith rucle,September1970,tobepublished, 32-1ustropucrion = Invecént yearstheFeynman rulesformasslessYang-Mills fieldshavebeenes- -Poe|‘blshed[1-5].Naivepowercountingsuggestsarenormalizable theory;however, [rrvaeod Bag,|©ordertocarrythrougharenormalization procedureonemustfirstdefineacut-off WalaonlyAwowLewp],sexoLev &Hucedure,Andifthecut-offprocedure breaksthegauge-invariance ofthetheory 44%4entisnomoreclearwhattheFeynmanrulesare,Thereasonisthatgavge- aSvariance,throughWardidentities,isessentialfortheS-matrixtobeunitary, Eg,|_Thustheproblemposesitseifasfollows:howtofindagaugeinvariantcut-off 24g"|Pocedure.Thisproblemisofcoursequitethesameinquantumelectrodynamics. *theretheproblem wassolvedbyPauli,Villars (6]andGupta [7]whosucceeded * Slindingasetofregulatorfieldsthatcouldbecoupledinagaugeinvariantway. TEE|SerithecaseofmasslessYang-Mills fieldsagaugeinvariantregularizing proce.Tybee,|7alsoseemstoexist.Unfortunately, however,thisprocedure cannotbeinter.‘tiedintermsoffieldswithindefinite metric and/or wrong statistics, likeinthea|ofelectrodynamics. Hence,unitarityandcausalityarenolongerevident “Mee.{However,itmustberealizedthatthewholerenormalization procedureinvolves Ay ia * “ 3 ~ yeMauria,ThehypermacleusSe 2CAT]Sete885G9717-18NormanbtnCony REFERENCESa [1]RELDattz, LeviSetti,Nuovo’Cimento30(196: i{2]Gohmeta,NookPhys.4G68)S11(1363)489, i . .Lemonneetal,Phys,Letters [5] RHStokesetal,Phys.Rev.178(1969)2024. (966). MASSIVE YANG-MILLS FIELDS. {6]J.Cernyetat,Phys,Rev.Letters16(1966)469, . (7]W.Gajewski etal,Nucl.Phys.37(1963)236 G..HOOFT 18]T.Lauritsen andFAjzenberg Selove,Nucl,Phys,78(1966)1. -GAHOOFT___ [9]Abetedoetal,NuovoCimento22(1961)LIT he InstitueforTheoretcelPhysics,UnversityofUtrechtJ.Lemonne etal,,NuovoCimento 34(1964)529, ¥ (10)P.H.Fowier andBH.Perkins, Phil,Mag,46(1958)587, s Received 13July1971 2|Abstract:Renornalizable modelszeconstructed inwhichloca!gageinvarianceisbroken& spontaneously. Feynman rulesandWardidentities canbefoundbymeansofapathin- 3‘egralmethod, andtheycanbechecked byalgebra, Inoneofthesemodels, whichis £studied inmore detail, local SUC) isbroken insuch away that local UC) remalns asa zsymmetry. Arenormalizable andunitary theory results, withphotons, charged massive . g vector particles, andaddtional neutral scalar particles, Ithas three independent param § cers. 3 ‘Another modelhaslocalSU(2)@U(I) asasymmetry andmayservea6arenormal- 3able theory formesons and photons. . Insuch model electromagnetic mase-differences arefinite and canbe calulated in 4 perturbation theory. Pa #1.INTRODUCTION % Inaprecedingarticle[1],henceforth referredtoasIithasbeenshownthat,ow-& ingtotheirlargesymmetry, mass-lessYang-Mills fieldsmayberenormalized, pro-& videdthatacertainsetofWardidentities isnotviolatedbyrenormalization effects,v2] Withhiswemeanthatanomalies likethoseoftheaxialcurrentWardidentities in . 29, nucleon-nucleon interactions (2~4],whichareduetoanunallowedshiftofinte- =} gration variables inthe“formal” proof, must notoccur. Initisproved thatsuch ¥ ‘anomalies areabsent indiagrams with oneclosed loop,iftherearenoparity- .|Ghangingtansformations ithelocalgagegroup.Wedoknowanextension ofthis"85; prootfordiagrams withanarbicary numberofclosed loops,buttsrathera Ef volved andweshallnotpresent ithere, By ‘Thus,ourprescription fortherenormalization procedure isconsistent, sotheze{ultraviolet problemformasslessYang-Millsfieldshasbeensolved,Amuchmore2g] complicated problemisformedbytheinfrareddivergencies ofthesystem.Wein- iberg[5]haspointed outthat,contrary tothequantum electrodynamical case,this 4 Problem cannot merely besolved bysome closer contemplation ofthemeasuring - process. Thedisaster issuchthattheperturbation expansion breaks down inthein- faredregion, sowehavenorigorous fieldtheory todescribe whathappens. EQUIVALENT ALTERNATIVES TOFADDEEV AND POPOV'S * : GAUGE THEORY QUANTIZATION PROCEDURE Carlos R.Handy ‘ Theoretical Division, Los Alamos Scientific Laboratory University ofCalifornia, LosAlamos, NewHexico 87545 ABSTRACT Analternate path integral gauge quantization procedure, developed ina preceding publication, isgeneralized. Two alternate quantization schemes can beformulated each equivalent tothat ofFaddeev and Popov. ‘The path integral representation foreither oftheseisoftheformJDA 6(GA)Ac{ A]exp(lE (ayaar), wheredfA}=(S{A}) Zac(D(A)G") forthe‘oriented’ procedure; or(MA})> + 1dee(o"'(a)G")) forthe'ionoriented' quantization ‘scheme. Md]istessentially' equal tothenumber ofnoncontinuous Gribov copies forA.S[A]isalsoaninte ger. The above isthe result ofananalysis independent ofgroup measure con- siderations, contrary toF-P's formalism. The existence ofGribov copies in theGigauge generally implies that agauge transformation from allofGWspace into GAspace, A[W] ,will not beinvertible. Thus afunctional change of variables-gauge transformation, from one gauge tothe other, which takes into account the noninvertibility ofthe mapping must beused. There are two ways ofrealizing this, the 'oriented' and 'nonoriented' prescriptions. Each leads . to Asofthe above form. Simple examples are given tocontrast the different - methods of gauge quantization. ‘ork supported bythe Department ofEnergy. yoy ae Volume 780,number1 PHYSICSLETTERS 1September1978 hen a ivy ies: ‘THERADIATION GAUGE STATICPOTENTIAL FORNON-ABELIAN GAUGE FIELDS a 1ae Samuel DAVISJeet LymanLaboratory, Department ofPhysics,HervardUniversity,Cambridge,MA02138,USA a andoo Frank L.FEINBERG?Sie CenterforTheoreticalPhysics,LaboratoryforNuclearScienceandDepartment ofPhysic,are Massachusetts InstituteofTechnology, Cambridge, MA02139,USA Received 16May 1978 Thestaticpotentialbetweenafermionandanantifermion inagroupsingletstateicalculated, throughtwoloops,in’ theradation gaugefirstorderformalism, Theresultsofthisealeulaton implythattheCoulomb propagator isnotsufficient todetermine thestatic potential: anewfunction ofthecoupling constant ag(~1) isalsorequired. Thenon-relativistic limitofthestronginteractions _seeninperturbation theorybelowthetwolooplevel, hasattractedconsiderable theoretical interestrecently _implythatitisproportional toa,(—1)multiplied bybecause ofitsrelevance fortheJ/yandnowtheT newdimensionless function ofa,(—t).systems [17]. Thespecific object ofinterest has Itisimportant tonotethatperturbative results beenthestaticpotential energy between averymas- ‘mustbeinterpreted withcaution. General features of sivefermion andaverymassive antifermion. Inthis thepotential, suchasthosedescribed above,maybeletterwereporttheresultsofanexplicit calculation extracted fromtheperturbative probeofthepotential,ofthestaticpotential between aheavyfermion and butthoserelationships whichdepend intrinsically on anti-fermion inthesinglet channel tosixthorderin thesmallness ofthecoupling constant cannot beas the'coupling constant fornon-abelian gaugefields sumedtobevalid,asmaybeillustrated bycomparison‘minimally coupledtothefermions. Althoughpertur- withQED.Naiveperturbation theorygivesinbothbation theory cannot beexpected toreproduce acon- theabelian andnon-abelian theoriesa~u/c,wherev finingpotential suchasthatfoundincharmonium, isthecharacteristic speedoftheconstituents ofthe itstillprovides someimportant information about boundstate.InQEDweexpect thisrelationship tobetheformofthepotential, Forexample, onlyinthe validbecause a<Iandbecause theexactnon-relati- singletsectordoesastaticpotential exist.Herewe visticpotential isgivenbythesingleCoulomb exchange.describe adetailed examination oftheperturbative Forthenon-abetian model, however, thenon-relativistic Structure ofthepotential whichrevealsthatthepoten- potential itselfisaninfinite powerseriesinthecouplingtialisnotdirectly proportional to@,(~£), thestsong constant,andonlythelowestordertermisgivenby couplingconstant. Additional dynamical effects, not _thesingleCoulomb exchange. Therefore, theremay exist bound states which arethreshold (with small ani i; Kineticenergyoftheconstituents) butwithoutany FoundationungatportedinartbytheNainaSconceSimpleconnectionbetweenufeandthestrongoovpling 2ThisworkssupportedInpanttvooghfundsprovided byconstant.Also,thepotentialneednotbecomestronger theUS.Department ofenergy(DOE)undercontract asthecouplingconstantincreases, unlikenon-relativisticEY-76-C-02-3069. QED. 90 \ss bse|abotwahdecays NiasPosi81421570157-196 Jade)156 Jobe Cortés et ©North-Holland Publishing Company [5]R.E,Marshak, Riazuddin andC.P,Ryan, Theory ofweakinteractions inparticle physics, (Wiley, New York, 1969). .. [6]M.Gronau, Phys, Rev.DS(1972) 118andreferences therein; .lag} D.Gomes,NucPhys.B130(1977)18, GiulowAw 1)1.F, Donoghue, Phys. Lett. 698 (1977) 437. . [9]S.J.BrodskyandJ.R.Primack,Ann.ofPhys.52(1969)315. BOUNDARY TERMSAND.POINCARE INVARIANCE * [9a]B.Kellet, Ann, ofPhys. 87(1974) 60; ‘LeYaouancetal.,Phys.Rev.D9ean)StinPh patticlephysics, ItzhakBARS*andFredericGREENseta, Relativistic quarksandSU(6),inPhenomenology ofparticlephysics n ; 10]LeYaouanc etal.,Phys. Lett. 72B (1977) 53. . [it]ADeRajGeorgandSLGato,ysRey.D12(1975)147, Received2May1978[12]G.AltarcliandL.Maiani,Phys.Lett$2B(1974)351.13]C.Schmidt, Phys. Lett. 668 (1977) 353, J.KatzandS.Tatur,Phys.Rev.D14(1976)2267. {0boundaryterms.ThePoincaréalgebrahasanamaliesandclvsennyIntieenay [15]Partcle Data Group, Rev. Mod. Phys. 48(1976) 1. Physical sector.Asimplemethod isproposed fordealingwiththnlvasntary suchen ~E§~0.as.x5 ~«inaHamiltonian formalism. ItFoundthatineraupepatewshavePhysically meaningful non-trivial asymptotic behavior relatedtuinstanton effetsthyPresents anobstacle toperturbation theory. 1,Introduction Previousinvestigations (1)ofquantumchromodyamies (QCD)intheaxialgauge ‘ASOhaveremained obscurewithrespecttoboundary conditions. Thepurposeo¢ thepresent paperistoclarifycertainproblems associated withasymptotic hehavioe, Wearemotivated byrecentwork[2]intwo-dimensional QCDwhichhasresvived .certaincomplications withPoincaré andgaugeinvariance insimilarcircumstances,Hereweinvestigate thenatureofanalogous problems aswellassomenewonesthatarisein4dimensions. Furthermore itisnowapparent thatthetraditional Coulombgaugecondition V-A*=0doesnotuniquely determine thepotentials innon-Abelian:gaugetheories [3].Thefactthattheaxialgaugedoesnotpresentthisproblem: provid:uswith afurther motivation. Thetheory isdefined bytheusualLagrangian LoRok +UGIB —my; Dy=OyigIAs, Fis*AS—Ad+gfARAL, (ay *ResearchSupported inpart(ValeReportNo.C00-3075-191) bytheUSDepartment o”Energy‘under Contract No.EY-76:C-02-3075. * Alfred P.Sloan Foundation Fellow. 187 i, “* i . PHYSICAL REVIEWD VOLUME10,NUMBER10 tswovenpen ys| Dynamical symmetry breaking ofnon-Abelian gaugesymmetrics*? et™potty: EstiaJ.Eichten scoot ferved € FrankL.Feinberg, +Syetiant DepartmentofPhysics, University ofCalifornia, LosAngles, California $0024 ere(Reeced1May197) joints WeusetheSchwinger mechanism togenerate adynamical breakdown ofnon-Abelian gauge 4Spical fsymmetries. Suchabreakdown isimplemented byusingbound-state Goldstone bosons which violate the erginéslobalinvarianceassociatedwiththegaugegroup.TheusefulnessofthitrealizationoftheSchwinger |ailar| imechanismjsthatiteliminates thenecessity ofintroducing elementary scalarparticles, and, sat furthermore,itis@viablepossibilty inapureYang-Mills theory.Fermion andvector-meson mass 2 ee, relationsinthepoleapprotimation arediscussed andcompared withthezeroth-order massrelations in we thefamiliar Higgsmodels. tnaddition, weobtainconsistent solutions totheBethe-Sslpeter equation for ve thebound stateintheweak-coupling limit,which yieldfinitevector-boson masses inthepole *compose approximation. However, therearegroup-theorctcal constraints which limitthepossible groups and .ratis,‘representation. Jiolating era mo} satewhi} 1.INTRODUCTION andthepoleinthepolarization tensor canbeseeg nanexplicitly, buttheGoldstone theorem isevaded sheAt Sincethesuccessof’tHooftandVeltman!andduetoanomalieswhichviolateaxial-vector conser.|¥denb: LeeandZinn-Sustn® inshowingtherenormaliza~ _vation,?Thereforesymmetryviolationleading|S#ée®Dittyofmassless Yang-MMis Lagrangians, there miassive particles isduetoanomalies, rathertua|BSe°™! hasbeenconsiderable interest’infieldtheories bound-state Goldstone bosons. anes Fospossessing localnon-Abelian gaugesymmetries ofMostofthecurrentresurgence ofinterestin«271.90 Eevector mesons. Hopehasbeenraisedthatweak spontaneous symmetry breakdown hasbeencene MESS.4 andelectromagnetic interactions maybeunitedbyteredaround2differentrealization oftheSchwin.|"NAPS= Usingthsclasoffieldtheories.?Morerecently, germechanism, thepopularHiggsmechaniem™ |FO!t Zl modelshavebeenintroducedwhichtrytoineor- JnthisprocedurescalarparticlesintheLagran-|$0070 =porate strong-interaction dynamics intothis gianareassigned symmetry-violating vacuum eevee ~scheme.Thekeyelementinthisapproachistheexpectation values,whichsupplytheraisond’étre|MOMS <{ntrodvetion ofspontaneous symmetry breaktows, forthepoleinpolarization tensor. However, |{'62°% 7sothatS-matrixelementsdonotexhibitallthe isdesirabletoproducemodelswithoutfundamen.|[0° <=formalsymmetrypropertiesoftheLagrangian, talscalarparticlesintheLagrangian becausete| >Excludinganomalies,weknowthatsuchcircum- sealarswhichareneededfortheHiggscasehave|A18°* aadstancesdemandtheappearance ofmassless Gold- _notbeenobserved, andfurthermore, itisdiffi- nese _stoneexcitations. However, sincetheLagran- culttoarrangeasymptotic freedomwhenscalar cess =giansareinvariant underlocalgaugetransforma- particles arepresent savation =tions,theGoldstoneexcitations combinewiththe ‘Twotechniques, mentionedabove,canbeused|Solu =originally massless vectormesons associated toavoidtheuseoftheconventional Higgs-Kibble j64s, =Withthegaugesymmetry toproducemassive vee- __mechanism"* inYang-Mills theories. One,which|POriurt rrtormesons. maybepeculiartotwodimensions," istoemploy|"0"!Sometimeago,Nambu andJona-Lasinio? in anomalies, whichdestroy current conservation, anthertheirpioneering workshowedthatatheorymay toprovidethepoteinthepolarization tensorofthe|(0ct-Possess symmetry-violating solutions byusinga vector particles. ‘Theotheristoassume thatthe eredhefour-Fermi interaction foproduceachiral-sym- integralequationsofthetheoryadmitsymmetry |{104Pemetry-breaking bound-state Goldstone boson. violation suchthettherearebound-state Gold~ nsec‘ThenextyearSchwinger® madethe’crucial obser- _stonebosonswhichgenerate thedesired pole. inthepvationthatitispossibletohave2poleinthepo- Usingthelattermethod,JackiwandJohnson’and|OFlarization tensor ofvector mesons at@"=0. Thts Cornwall andNorton,:" extending theworkof benePoleleadstospontaneous symmetry breakdown by Nambu andJona-Lasinio,” showed thatitiscon= neonsivingamass”tootherwise massless vector me-_sistent foragaugetheory offermions interact- |3"¢£2" sons,@processknownastheSchwinger mechan- ingwithmassless Abelianvectormesonstohave|.ction ism. InSchwinger’ sexample,* two-dimensional symmetry-breaking solutions such thatthere is2honete quantumelectrodynamics, themodel issoluble finitephysical massforthefermion andforthe inthes| forpur JO3254 7 re - ie!PRE 22014 ery CcJuESfy ‘ . \ kot Y,NOTTS > THE GLUON PROPAGATOR R,Delbourgo Department of ‘Physics, University of Tasmania, Hobart, Tasmania November 1978 , ABSTRACT ‘The vector giuon propagator ofquantum chromodynamics isinvestigated bymeans ofthe gauge technique. The technique provides aself consistency equation for the propagator from which one may recover the asymptotic freedom behaviour pA +[1+ &n(-p*/u*)]}+? inthe ultraviolet regime: p?>>y?. The corresponding behaviour inthe infrared limit p*<<uy? isfound tobe ped %[8n(~p?/u?)]-7. rf ~ 2 le INTRODUCTION Itisbecoming increasingly evident (Field, 1978) that perturbative computations inquantum chromodynamics (QCD) using effective coupling constants give reasonable agreement with the experimental data for large momenta. The ability toperform and believe such calculations ispredicated onthe phenomenon ofasymptotic freedom (Gross andWilezek, 1973; Politzer, 1973) fortheultra~ violet regime whereby the effective coupling constant g(p*), for p*+=, : behaves as - 37(7)%go?(-u2)+cgfn(-p*/u?) foo) with c,,>0 and urepresenting arenormalisation mass scale. Anequivalent: restatement isthat the gluon propagator has the high energy behaviour Ap?) &pr?LL+8?(-u2)an(~p2/u2) J? (2) : What happens inthe infrared to g(p*) or A(p?) ismuch less clear and has become the subject ofsome debate. Awidespread conjecture isthat . as p*+0, A(p?) ~p-* inorder forcolour confinement totake place bya linear potential orcolour flux tube. Largely because ofthe quark confinement question, asubstantial amount oftheoretical research has gone into investigating the infrared characteristics of nonabelian theories (Cornwall and Tiktopoulos, 1976; Frenkel etal., 1977), both inaperturbative framework (Appelquist etal., 1976; Konetschny, 1978) -where, according tothe order of summation, new effects may ormay not show up~and also onthe basis ofnon- perturbative techniques (Pagels, 1977). Asyet nodefinitive, universally - accepted result has emerged from these various attempts.